# 2+2 is 4 and the world is flat: assumptions and approximations

The argument that “2+2=4” is a social construct not found in every society and that in some places 2+2=5 is an interesting exercise in sophistry.

It is true that you can redefine every part of an equation (or every word in a sentence) to mean something other than what a naive reader would normally assume based on all previous experience with words. Obviously if we redefine 2 to mean something other than 2, + to mean something other than addition, or chose to use a base other than base 10, then we can get an answer other than 4. For example, if we are using base-3, then 2+2 = 11. (Of course, when we convert back to base ten, “11” becomes plain old 4 again.)

This is true of every sentence: if I redefine all of the words in a sentence to mean something else, then the sentence means something else–but no one uses language in this way because it makes communication impossible.

In particular, when children are taught that 2+2=4, they being taught within a system where 2 means two of something, 4 means four of something, and + means conventional addition. When we use these definitions, 2 + 2 always equals 4. There are, in fact, zero societies on Earth where this equation, as used in elementary schools, comes out to five.

There is no mistake involved in assuming that other people are using common conventions when speaking and will specify when using terms in unexpected ways. This is how all communication works. Since we cannot define all words from first principles every time we use them (this is impossible because it would require us to define the words used to define the words used to define the words, etc,) we only bother to define them when using them in unconventional ways, and even then we use conventionally defined words to define them. If a word is not defined or otherwise marked as being used in unconventional ways, then the receiver assumes that it is being used in its conventional sense because language cannot function otherwise.

Behind the scenes trickery is simply that: trickery. The sentence as normally defined and automatically understood is always correct.

There is a story about the time Denis Diderot visited the court of Catherine the Great in Russia. Diderot’s atheism offended the great lady, so she had her court mathematician, Leonhard Euler, confront Diderot with a complicated algebraic equation, then proclaimed that this proved the existence of God.

The tale is perhaps apocryphal, but it has inspired the coining of the term “Eulering”: the use of a complicated argument to confuse your opponent into conceding, especially on some irrelevant point. Modern Eulering often consists of saying something that sounds blatantly false, then when this is pointed out, ridiculing your opponent for not knowing that you had secretly redefined the words. If you were a real expert, the argument goes, then you would know that 10=2 is just as valid as 10=10, because base ten is just a social convention we use to make writing numbers easier, and all other bases–including base pi–are equally legitimate from a mathematical perspective.

This is not expertise, but sophistry. There is no mistake involved in assuming that other people are using common conventions when speaking and will specify when using terms in unexpected ways. This is how all communication works.

It is true that math as taught to children is simplified: all subjects are simplified by necessity for introductory students.

When a child learns to read, he is first taught to pronounce the letters phonetically; complications like “silent e” and “-tion” are only introduced later. The full complexity of English spelling, from rhythm to pterodactyl, is only revealed to advanced students who have already mastered simpler words. If we attempt to reverse the order of instruction, chaos results: students are forced to learn every single word independently, instead of applying general rules that get them through most of the words and help them develop further rules for the exceptions.

The same happens in math; children are taught to count and add with the help of simplifying assumptions like “triangles are flat” and “base 10.” You don’t teach a toddler to count by beginning with -10 and then explaining that “3” is written as “11” in base two. It doesn’t work.

When you learn physics, you begin with Newtonian dynamics, because these are easy to demonstrate at normal human scales. It is only after mastering the basics of F=ma, objects falling at 9.8 m/s^2, and maybe a bit of calculus that you move on to topics like “What happens when you move close to the speed of light?” or “What happens at the atomic scale?”

Subjects are taught in a particular order that equips students with general rules that work in most situations, then specific rules that cover the most common exceptions. Most people will never need to know the “expert level” versions of most fields. For example, most people do not need to understand why airplanes can fly in order to make a reservation at the airport and go on a trip: it is sufficient to know simply that planes fly.

To argue about whether the “basic” or “expert” versions of these fields is more correct  generally misses the point: each serves a specific purpose. If I am calculating the distance between my house and my friend’s, I do not need to factor in the curvature of the Earth; if I am calculating the distance between my house and the antipodes, I do. If I am balancing my checkbook, I can safely assume that all of the numbers are written in base-10; if I am trying to figure out if the 16-bit integer limit will make my airplane crash, then it helps to know binary. If I tell my kids to “stay still so I can take your picture,” I don’t want to hear that Brownian motion technically makes it impossible to hold still.

The current bruhaha on Twitter over whether “2+2=4” is racist or not is half math geeks happy to finally have an audience for their discussion of obscure math things and half “school reformers” who wouldn’t know ring addition if it hit them in the face but want to claim that it has something to do with early elementary math. (Spoiler: it doesn’t.)

In the case of math, yes, math is a social construct, inasmuch as we could use a different numerical base or different wiggly symbols to represent the numbers on paper. Spelling is also a social construct: there is no particular reason why “C” should be pronounced the way it is, much less should we have a silent “L” in “could” (the L in could is actually the result of a centuries-old spelling mistake: “would” and “should” both contain Ls because they are forms of the words “will” and “shall,” which contain Ls. Could is derived from the word “can,” which does not have an L, but because “coud” sounds like “would” and “should,” people just started sticking an erroneous L in there, and we’ve been doing it for so long that it’s stuck). Money is also socially constructed: there is no particular reason why little green pieces of paper should have any value, and in many cases (lost in the woods, hyperinflation, visiting a foreign country), they don’t. Nevertheless, you need to be able to count, spell, and use money to get along in society, which we live in. If math is racist simply because it is socially constructed, then so are all other social constructs. Pennies are racist. Silent “e” is racist.

This absurdity is no accident:

h/t @hollymathnerd, quote by Shraddha Shirude, “ethnomathematics” teacher.

# Noise, Noise, Noise

Noise is a the foe of any information transmission, yet the total elimination of noise is–counter-intuitively–bad.

Nature is, from our human perspective, inherently noisy, and this noise is inherent to its beauty. When we try to over-fit a simple order we get something that is, yes, lined up neatly, but also dead.

Nature is not truly noisy, but the mathematics that underlies its order is more complicated than we can easily model (it is only in the past century that we’ve developed the tools necessary to model tomorrow’s weather with any degree of accuracy, for example.) Whether we are drawing trees or clouds, controlled randomness works far better than regular repetition. (If you want to get technical, the math involved tends to involve fractals.)

In photography, dithering is the intentional application of noise to randomize quantitization errors. According to Wikipedia:

Dither is routinely used in processing of both digital audio and video data, and is often one of the last stages of mastering audio to a CD.

A common use of dither is converting a greyscale image to black and white, such that the density of black dots in the new image approximates the average grey level in the original.

…[O]ne of the earliest [applications] of dither came in World War II. Airplane bombers used mechanical computers to perform navigation and bomb trajectory calculations. Curiously, these computers (boxes filled with hundreds of gears and cogs) performed more accurately when flying on board the aircraft, and less well on ground. Engineers realized that the vibration from the aircraft reduced the error from sticky moving parts. Instead of moving in short jerks, they moved more continuously. Small vibrating motors were built into the computers, and their vibration was called dither from the Middle English verb “didderen,” meaning “to tremble.” Today, when you tap a mechanical meter to increase its accuracy, you are applying dither, and modern dictionaries define dither as a highly nervous, confused, or agitated state. In minute quantities, dither successfully makes a digitization system a little more analog in the good sense of the word.

— Ken Pohlmann, Principles of Digital Audio[1]

The term dither was published in books on analog computation and hydraulically controlled guns shortly after World War II.[2][3]

This mechanical dithering is also why many people play “white noise” sounds while studying or falling asleep (my eldest used to fall asleep to the sound of the shower). As the article notes:

Quantization yields error. If that error is correlated to the signal, the result is potentially cyclical or predictable. In some fields, especially where the receptor is sensitive to such artifacts, cyclical errors yield undesirable artifacts. In these fields introducing dither converts the error to random noise. The field of audio is a primary example of this. The human ear functions much like a Fourier transform, wherein it hears individual frequencies.[8][9] The ear is therefore very sensitive to distortion, or additional frequency content, but far less sensitive to additional random noise at all frequencies such as found in a dithered signal.[10]

In digital audio, like CDs, dithering reduces the distortion caused by data compression (a necessary part of the process of producing CDs). The same process works in digital photography, as demonstrated in these photos:

In art, dithering allows artists to create a wide range of colors out of a limited palette (this is, of course, how the colors in newspaper comics are created).

There are many different ways to dither, and subsequently many different dithering algorithms, all created with the intention of using random noise to increase the quality of images/sound/readings, etc.

Too much noise is of course a problem, but as photographer Frederik Mork notes:

Don’t forget that noise can also be a creative tool. Especially when paired with black-and-white, high ISO noise can sometimes add a lot of atmosphere to the image. Some images need noise.

Symmetrical faces are supposed to be more attractive than less-symmetrical faces, but this only applies up to a point. Even very beautiful faces are not truly symmetrical, and increasing their symmetricality (by mirroring the left side over the right, or the right side over the left) does not improve them:

I don’t know where this image came from originally, but I found it in a YouTube video.  There is a literature on this subject, probably primarily related to plastic surgery. The image on the left is a Nicole as she normally appears; the image in the center is Nicole with the right side of her face mirrored, an the image on the right is the left side of her face. Note that the original contains several asymmetries: her hair, her eyebrows, and even the way her necklace lies on her chest.

This post was inspired by a conversation about what it meant to be a “good” musician, and whether one can be a reasonable judge of music outside of one’s own musical tastes. If I love rap but hate techno, can I still recognize which techno songs are considered “good” by some standard other than “techno fans like it”? Is all art subjective, or are some pieces actually better or worse than others?

I am a simple creature, and I do not really understand the depths of what goes into creating music. I can squeak out a few notes on a recorder and Twinkle Twinkle Little Star on the piano, but deeper theories behind things like “harmonic intervals” and “chords” are beyond me. Music to me is an immersive sensory experience (more so since quarantine has induced a state of semi-zen that has quieted the normally very loud meta and meta-meta narrative in my head). Whether something is good or bad I cannot say in any quantitative sense, but to paraphrase the words of Justice Stewart, I know it when I feel it.

I had very little exposure to popular music as a kid (I listened to a lot of “Christian rock”), and only began listening to music seriously and sorting out what I liked and didn’t in grad school, so whatever arguments you have about people forming their musical taste in highschool don’t apply.

I tend to like music that has a lot of distortion–noise, if you will. (I like music that reflect what I feel, and my feelings look like a Jackson Pollock.)

You might have noticed that my current favorite musician is Gary Numan (the guy who sang “Cars” back in 1979.) I don’t like most of Numan’s early work (like “Cars”); the sound is too simple. It took a couple of decades for Numan to develop the layers of complexity and necessary to create the sort of musical soundscape I enjoy. Compare, for example, his 1979 performance of “Are Friends Electric” to his 2013 performance of the same song. They are the same song, but with slightly different arrangements; I don’t know the technical words to describe them, but I find the 1979 version merely acceptable, while the 2013 hit me like a brick. (That’s a good thing, in this context.)

Or, heck, let’s compare Cars 1979, with Cars 2009 (with NIN). Okay, so the first thing that stands out to me is that 2009 Gary Numan has gotten laid and is no longer afraid to move around the stage, unlike 1979 Numan. Second, Trent Reznor on the tambourine is hilarious. Third, there is WAY more noise in the second recording–especially since it is live and there is a screaming audience–and this does not detract from the experience: I vastly prefer the 2009 version.

One of the things I love about Numan’s music is that you can do this; you can listen to the same piece from different eras and see how his style has changed and evolved.

To really appreciate his new new style, though, I think it’s best to listen to new compositions, rather than covers of his older work, like I am Dust, Ghost Nation, or Crazier (with Rico).

Music, as I understand it, is built from a disturbed mathematical progression of sounds. A sequence of sounds in a regular pattern builds up our expectation of what will come next, and the violation of this sequence creates surprise, which–when done properly–our brains enjoy. If the pattern simply repeated over and over, it would become boring.

I recently enjoyed a documentary on Netflix about ZZ Top, (who knows, maybe they’ll become my favorite band someday). At one point in the documentary the band described the difficulties of their first recording session. They had their song, had their band, had their instruments, but the guy doing the recording just couldn’t capture the right sound. He had microphones all around the recording studio, but just couldn’t get what he wanted. So he proposed that the band loosen the strings on their guitars (or maybe it was just one guitar, forgive me, it’s been a while) to create a slightly out of tune sound. The band’s manager was having nothing of it: the instruments needed to be in tune. Finally the recording studio guy proposed that the manager run out to get them some barbecue, because it was getting on toward dinner time, and conveniently directed the manager to a restaurant across the county line, a good half hour away. With the manager gone for the next hour and fifteen minutes, they untuned the guitars and finally got the sound they were looking for.

(Here’s ZZ Top’s Sharp Dressed Man.)

Is this a “better” sound? It’s a different sound. It’s not the standard sound, and if you’re looking for the standard sound that guitars are supposed to make, this isn’t it. Is it wrong, on technical grounds? Or is it right because it was the sound the band wanted to make?

This whole post was inspired by the claim that Kurt Cobain was, technically, not a good musician. I found this accusation absolutely flabbergasting. Of course, I regard pretty much anyone who can crank out a tune on a guitar as “good”, and anyone who can top the charts as “excellent” by default. If we want to differentiate between top stars, well, I guess we can, but in the immortal words of @dog_rates, they’re all good dogs, Brent.

I’m not a huge Nirvana fan–like I said, I never listened to them as a kid (I didn’t have MTV and couldn’t have told you the difference between Pearl Jam and Oyster Jelly), but I like grunge and Nirvana is part of that.

After some conversation, I realized that my interlocutor and I were using different definitions of “technically,” which is always a sign that you should stop arguing about dumb stuff on the internet and go take a walk, except when you’re quarantined. I meant “technically” as in “actually,” while he meant it in the sense of “the proper way of doing something; by the manual.” His argument was that Kurt Cobain did not play the guitar properly according to the manual for how to play a guitar. Kurt was dropping notes, or not pushing down the strings properly, or otherwise playing lazily and not doing it right. Someone who has taken lessons on “how to play a guitar” will not be able to play like Kurt because he was being sloppy and not playing properly. They will have to learn how to be sloppy.

Naturally, I found this argument baffling. I have no idea how to play the guitar, but my standard for whether a piece of music is good or not is based on how it sounds, not how it is produced. I am listening to Nirvana now and I don’t hear–to my ears–any flaws.

To say that there is a “proper” way to play a guitar that is a standard benchmark against which musicians are judged sounds like some sort of  prescriptivist nonsense. It’s like saying that there is a proper way to paint, and that “impressionism” isn’t it:

An outraged critic, Louis Leroy, coined the label “Impressionist.” He looked at Monet’s Impression Sunrise, the artist’s sensory response to a harbor at dawn, painted with sketchy brushstrokes. “Impression!” the journalist snorted. “Wallpaper in its embryonic state is more finished!” Within a year, the name Impressionism was an accepted term in the art world.

If the name was accepted, the art itself was not. “Try to make Monsieur Pissarro understand that trees are not violet; that the sky is not the color of fresh butter…and that no sensible human being could countenance such aberrations…try to explain to Monsieur Renoir that a woman’s torso is not a mass of decomposing flesh with those purplish-green stains,” wrote art critic Albert Wolff after the second Impressionist exhibition.

Although some people appreciated the new paintings, many did not. The critics and the public agreed the Impressionists couldn’t draw and their colors were considered vulgar. Their compositions were strange. Their short, slapdash brushstrokes made their paintings practically illegible. Why didn’t these artists take the time to finish their canvases, viewers wondered?

Indeed, Impressionism broke every rule of the French Academy of Fine Arts, the conservative school that had dominated art training and taste since 1648.

The “proper” way to play a guitar is however sounds good, and if it sounds better with dropped notes and imperfectly depressed strings, then that is the proper way to play. This is grunge, and grunge is intentionally high on the distortion.

Whether it sounds good or not is, of course, a matter of opinion, but Smells Like Teen Spirit has over a billion views on YouTube, so I think it’s fair to say that there are a lot of people out there who think this is a very good song. From their perspective, Kurt Cobain is a very talented musician.

There’s a saying that you have to learn the rules in order to know when to break the rules. It applies primarily in art and literature, but I suppose it applies to the rest of life, too. When you are learning to write, you learn the rules of grammar and punctuation. If you become a poet, you know when and how to throw all of that out the window. If you want to be an artist, you need to know how to paint; later you can throw together whatever colors you want. If you want to play music, then you need to learn to play properly–harmonic intervals and all that, I suppose–but when you actually play music, you need to know when to break the rules and how subtle differences in the way the notes are played result in massive differences in the music. Compare, for example, NIN’s Hurt to Johnny Cash’s.

This is a song that gets its power from our subverted expectations; we expect an increase in tempo that never really comes, creating a tension that stretches out across the song, finally breaking at 4:33 (in Trent’s version).

In these two songs, I think Trent’s version is “better” in the technical skills sense, but Cash’s version is better in the absolute punch in the guts sense. This is simply because of who Trent and Cash are; they each bring their own sense of self to the song: Trent the sense of a bitter youth; Cash the sense of an old man composing his epitaph.

Let us end with some Alice in Chains:

I hope you have enjoyed the songs.

# A hopeful note on ability distributions

I read recently (my apologies, I can’t find the link) that in every country where we have reliable testing data, a consistent pattern emerges: girls tend to do slightly better on reading/writing tasks than mathematical tasks, and boys slightly better on mathematical than language-tasks.

This is an interesting dynamic because it creates different “optimal” outcomes depending on what you are trying to optimize for.

If you optimize for individual achievement–that is, get each student to go into the field where they, personally, can do the best–the vast majority of girls will go into language-related fields and the vast majority of boys will go into math-based fields. This leaves us with a strongly gender-divided workforce.

But if we optimize instead for getting talented people into a particular field, the gender divide would be narrower. Most smart students are good at both math and language, and could excel in either domain. You could easily have a case where the best mathematician in a class is even more talented in language, or where the most verbally talented person is even more talented at mathematical tasks (but not both at once).

If we let people chose the careers that best suit them, some fields may end up sub-optimally filled because talented people go elsewhere. If we push people into particular fields, some people will end up sub-optimally employed, because they could have done a better job elsewhere.

Relatedly, we find that people show more gendered job preferences in developed countries, and less gendered preferences in undeveloped countries. In Norway, women show a pretty strong preference, on average, for careers involving people or language skills, while in the third world, they show a stronger preference for “masculine” jobs involving math, science, or technical skills. This finding is potentially explained by different countries offering different job opportunities. In Norway, there are lots of cushy jobs, and people feel comfortable pursuing whatever makes them happy or they’re good at. In the third world, technical skills are valued and thus these jobs pay well and people strive to get them.

People often ascribe the gender balance in different jobs to nefarious social forces (ie, sexism,) but it is possible that they are an entirely mundane side effect of people just having the wealth and opportunity to pursue careers in the things they are best at.

# The Female Problem

As Pumpkin Person reports, 96% of people with math IQs over 154 are male (at least in the early 1980s.) Quoting from  Feingold, A. (1988). Cognitive gender differences are disappearing. American Psychologist, 43(2), 95-103:

When the examinees from the two test administrations were combined, 96% of 99 scores of 800 (the highest possible scaled score), 90% of 433 scores in the 780-790 range, 81% of 1479 scores between 750 and 770, and 56% of 3,768 scores of 600 were earned by boys.

The linked article notes that this was an improvement over the previous gender gap in high-end math scores. (This improvement may itself be an illusion, due to the immigration of smarter Asians rather than any narrowing of the gap among locals.)

I don’t know what the slant is among folks with 800s on the verbal sub-test, though it is probably less–far more published authors and journalists are male than top mathematicians are female. (Language is a much older human skill than math, and we seem to have a corresponding easier time with it.) ETA: I found some data. Verbal is split nearly 50/50 across the board; the short-lived essay had a female bias. Since the 90s, the male:female ratio for scores over 700 improved from 13:1 to 4:1; there’s more randomness in the data for 800s, but the ratio is consistently more male-dominated.

High SAT (or any other sort of) scores is isolating. A person with a combined score between 950 and 1150 (on recent tests) falls comfortably into the middle of the range; most people have scores near them. A person with a score above 1350 is in the 90th%–that is, 90% of people have scores lower than theirs.

People with scores that round up to 1600 are above the 99th%. Over 99% of people have lower scores than they do.

And if on top of that you are a female with a math score above 750, you’re now a minority within a minority–75% or more of the tiny sliver of people at your level are likely to be male.

Obviously the exact details change over time–the SAT is periodically re-normed and revised–and of course no one makes friends by pulling out their SAT scores and nixing anyone with worse results.

But the general point holds true, regardless of our adjustments, because people bond with folks who think similarly to themselves, have similar interests, or are classmates/coworkers–and if you are a female with high math abilities, you know well that your environment is heavily male.

This is not so bad if you are at a point in your life when you are looking for someone to date and want to be around lots of men (in fact, it can be quite pleasant.) It becomes a problem when you are past that point, and looking for fellow women to converse with. Married women with children, for example, do not typically associate in groups that are 90% male–nor should they, for good reasons I can explain in depth if you want me to.

A few months ago, a young woman named Kathleen Rebecca Forth committed suicide. I didn’t know Forth, but she was a nerd, and nerds are my tribe.

She was an effective altruist who specialized in understanding people through the application of rationality techniques. She was in the process of becoming a data scientist so that she could earn the money she needed to dedicate her life to charity.

I cannot judge the objective truth of Forth’s suicide letter, because I don’t know her nor any of the people in her particular communities. I have very little experience with life as a single person, having had the good luck to marry young. Nevertheless, Forth is dead.

At the risk of oversimplifying the complex motivations for Forth’s death, she was desperately alone and felt like she had no one to protect her. She wanted friends, but was instead surrounded by men who wanted to mate with her (with or without her consent.) Normal people can solve this problem by simply hanging out with more women. This is much harder for nerds:

Rationality and effective altruism are the loves of my life. They are who I am.

I also love programming. Programming is part of who I am.

I could leave rationality, effective altruism and programming to escape the male-dominated environments that increase my sexual violence risk so much. The trouble is, I wouldn’t be myself. I would have to act like someone else all day.

Imagine leaving everything you’re interested in, and all the social groups where people have something in common with you. You’d be socially isolated. You’d be constantly pretending to enjoy work you don’t like, to enjoy activities you’re not interested in, to bond with people who don’t understand you, trying to be close to people you don’t relate to… What kind of life is that? …

Before I found this place, my life was utterly unengaging. No one was interested in talking about the same things. I was actually trying to talk about rationality and effective altruism for years before I found this place, and was referred into it because of that!

My life was tedious and very lonely. I never want to go back to that again. Being outside this network felt like being dead inside my own skin.

Why Forth could not effectively change the way she interacted with men in order to decrease the sexual interest she received from them, I do not know–it is perhaps unknowable–but I think her life would not have ended had she been married.

A couple of years ago, I met someone who initiated a form of attraction I’d never experienced before. I was upset because of a sex offender and wanted to be protected. For months, I desperately wanted this person to protect me. My mind screamed for it every day. My survival instincts told me I needed to be in their territory. This went on for months. I fantasized about throwing myself at them, and even obeying them, because they protected me in the fantasy.

That is very strange for me because I had never felt that way about anyone. Obedience? How? That seemed so senseless.

Look, no one is smart in all ways at once. We all have our blind spots. Forth’s blind spot was this thing called “marriage.” It is perhaps also a blind spot for most of the people around her–especially this one. She should not be condemned for not being perfect, any more than the rest of us.

But we can still conclude that she was desperately lonely for normal things that normal people seek–friendship, love, marriage–and her difficulties hailed in part from the fact that her environment was 90% male. She had no group of like-minded females to bond with and seek advice and feedback from.

Forth’s death prompted me to create The Female Side, an open thread for any female readers of this blog, along with a Slack-based discussion group. (The invite is in the comments over on the Female Side.) You don’t have to be alone. (You don’t even have to be good at math.) We are rare, but we are out here.

(Note: anyone can feel free to treat any thread as an Open Thread, and some folks prefer to post over on the About page.)

Given all of this, why don’t I embrace efforts to get more women into STEM? Why do I find these efforts repulsive, and accept the heavily male-dominated landscape? Wouldn’t it be in my self-interest to attract more women to STEM and convince people, generally, that women are talented at such endeavors?

I would love it if more women were genuinely interested in STEM. I am also grateful to pioneers like Marie Curie and Lise Meitner, whose brilliance and dedication forced open the doors of academies that had formerly been entirely closed to women.

The difficulty is that genuine interest in STEM is rare, and even rarer in women. The over-representation of men at both the high and low ends of mathematical abilities is most likely due to biological causes that even a perfect society that removes all gender-based discrimination and biases cannot eliminate.

It does not benefit me one bit if STEM gets flooded with women who are not nerds. That is just normies invading and taking over my territory. It’s middle school all over again.

If your idea of “getting girls interested in STEM” includes makeup kits and spa masks, I posit that you have no idea what you’re talking about, you’re appropriating my culture, and you can fuck off.

Please take a moment to appreciate just how terrible this “Project Mc2” “Lip Balm Lab” is. I am not sure I have words sufficient to describe how much I hate this thing and its entire line, but let me try to summarize:

There’s nothing inherently wrong with lib balm. The invention of makeup that isn’t full of lead and toxic chemicals was a real boon to women. There are, in fact, scientists at work at makeup companies, devoted to inventing new shades of eye shadow, quicker-drying nail polish, less toxic lipstick, etc.

And… wearing makeup is incredibly normative for women. Little girls play at wearing makeup. Obtaining your first adult makeup and learning how to apply it is practically a rite of passage for young teens. Most adult women love makeup and wear it every day.

Except:

Nerd women.

Female nerds just aren’t into makeup.

I’m not saying they never wear makeup–there’s even a significant subculture of people who enjoy cosplay/historical re-enactment and construct elaborate costumes, including makeup–but most of us don’t. Much like male nerds, we prioritize comfort and functionality in the things covering our bodies, not fashion trends.

And if anything, makeup is one of the most obvious shibboleths that distinguishes between nerd females and normies.

In other words, they took the tribal marker of the people who made fun of us throughout elementary and highschool and repackaged it as “Science!” in an effort to get more normies into STEM, and I’m supposed to be happy about this?!

I am not ashamed of the fact that women are rarer than men at the highest levels of math abilities. Women are also rarer than men at the lowest levels of math abilities. I feel no need to cram people into disciplines they aren’t actually interested in just so we can have equal numbers of people in each–we don’t need equal numbers of men and women in construction work, plumbing, electrical engineering, long-haul trucking, nursing, teaching, childcare, etc.

It’s okay for men and women to enjoy different things–on average–and it’s also okay for some people to have unusual talents or interests.

It’s okay to be you.

(I mean, unless you’re a murderer or something. Then don’t be you.)

# The World is Written in Beautiful Maths

This is a timelapse multiple exposure photo of an arctic day, apparently titled “Six Suns” (even though there are 8 in the picture?) With credit to Circosatabolarc for posting the photo on Twitter, where I saw it. Photo by taken by Donald MacMillan of the Crocker Land Expedition, 1913-1917.

Attempting to resolve the name-suns discrepancy, I searched for “Six Suns” and found this photo, also taken by Donald MacMillan, from The Peary-MacMillian Arctic Museum, which actually shows six suns.

I hearby dub this photo “Eight Suns.”

A reverse image search turned up one more similar photo, a postcard titled “Midnight Sun and Moon,” taken at Fort McMurray on the Arctic Coast, sometime before 1943.

As you can see, above the arctic circle, the sun’s arc lies so low relative to the horizon that it appears to move horizontally across the sky. If you extended the photograph into a time-lapse movie, taken at the North Pole, you’d see the sun spiral upward from the Spring Equinox until it reaches 23.5 degrees above the horizon–about a quarter of the way to the top–on the Summer Solstice, and then spiral back down until the Fall Equinox, when it slips below the horizon for the rest of the year.

In other news, here’s a graph of size vs speed for three different classes of animals–flying, running, and swimming creatures–all of which show the same shape. “A general scaling law reveals why the largest animals are not the fastest” H/T NatureEcoEvo

I love this graph; it is a beautiful demonstration of the mathematics underlying bodily shape and design, not just for one class of animals, but for all of us. It is a rule that applies to all moving creatures, despite the fact that running, flying, and swimming are such different activities.

I assume similar scaling laws apply to mechanical and aggregate systems, as well.

# Homeschooling Corner: Summer Fun

Hello, everyone. I hope you have had a lovely summer. We ended up scaling back a bit on our regular schedule, doing about half as much formal “schoolwork” as usual and twice as much riding bikes and going to the playground.

Here are some of the books we found particularly useful/enjoyable this summer:

String, Straightedge, and Shadow: The Story of Geometry, by Julia E Diggins

This is my favorite book we read this summer.

I was looking for a book to introduce simple geometry and shape construction. Instead, I found this delightful history of geometry. It is appropriate for children who understand simple fractions, ratios, and the Pythagorean theorem, but it is not a mathematics textbook and only contains a few equations. (I’m still looking for an introduction to geometry, if anyone has any recommendations.)

This is a new edition of a book originally published in 1965, but its age isn’t really important because geometry hasn’t changed much in the intervening years.

The story begins with geometry in nature–the shapes of trees and flowers, spiderwebs and honeycombs–then develops a speculative account of how early stoneage humans might have become increasingly aware of and attuned to these shapes. Men saw the shapes of the sun and moon in the sky, and might have observed that an ox tied to a pole traced out a similar shape in the dirt.

Then Egyptian surveyors developed right triangles, used for measuring the corner of fields and pyramids. The Mesopotamians developed astronomy, and divided the circle into 360 degrees. Then came the Greeks–clever Thales, mystical Pythagoras, and practical Archimedes. And finally, at the end, Eratosthenes (who used geometry–literally, earth measuring–to measure the circumference of the Earth,) and a few paragraphs about Euclid.

Writing with Ease, by Susan Wise Bauer

There are many books and workbooks in this series, so you can pick the ones that best suit your child’s ability level. (The “look inside the book” feature is great for judging which level of textbook you want.)

I am sure these books are not everyone’s cup of tea. They may not be yours. But they were what we needed.

My eldest children are fairly different in writing needs, but I do not have time for separate curricula. One is a good speller, the other bad. One has acceptable handwriting, the other awful. One will write independently, the other hates writing and plays dead if I try to get them to write. These books have worked well for everyone. Spelling, handwriting, and general willingness to write have all improved.

Even if you aren’t homeschooling, this book might make a good supplement to your kids’ regular curriculum.

In science, we have been growing bacteria in petri dishes and looking at them under the microscope, with the help of Usborne Science and Experiments: The World of the Microscope (I think this is the same book, but cheaper.)

Petri dishes are cheap, agar is easy to make at home (it’s just like making jello,) and kids can learn things like “doorknobs are dirty” and “that’s why mom makes me wash my hands before dinner.”

Just be careful when handling large quantities of bacteria. Even if it’s normal household bacteria that you’re exposed to regularly, you’re not used to it in these quantities. The instructions recommend wearing gloves and safety goggles while handling bacteria and making slides out of them–and besides, kids like dressing up “like scientists” anyway.

Our pattern blocks have been in the family for decades–passed down to me by my grandmother–but the geoboards are a new acquisition. I remember geoboards in elementary school–they sat behind the teacher’s desk and we never actually used them. I didn’t know what, exactly, geoboards were for, so I went ahead and got new workbooks for both them and the pattern blocks.

We are only a few lessons in, but so far I am very pleased with these. We have been talking about angles and measuring the degrees in different shapes with the pattern blocks–360 in a circle, 180 in a triangle, 720 in a hexagon, etc–which dovetails nicely with the geometry reading. The geoboards let us construct and examine a variety of different shapes, like right and equilateral triangles. The lesson plans are easy to use and the kids really enjoy them. Just watch out for rubber bands flying across the room.

Super Source makes workbooks for different grade levels, from K through 6th.

Learn to Program with Minecraft, by Craig Richardson

This book introduces Python, and is a nice step up from the Scratch workbooks. You may have to install a couple of programs, like Python and the API spigot, but the book walks you through this and it is not bad at all. There are then step-by-step instructions for making simple programs, along with bonus challenges to work out on your (or your kid’s) own.

The book covers strings, booleans, if statements, loops, etc, in kid-friendly ways. Best for people who already love Minecraft and can type.

# Book Club: The Code Economy chs. 3-4: Machines and Computers

Machines are fascinating.

Imagine if we discovered, by some chance, in a previously unexplored niche of the world, a group of “people” who looked exactly like us, but had no technology at all: no fire, no pots, no textiles, not even an Acheulean hand axe. Let us further assume that they were not merely feral children who’d gotten lost in the woods, but an actual community that had sustained itself for generations, and that all attempts to introduce them to the art of tool-making failed. They could not, despite often watching others build fires or throw pots, make their own–much less understand and join in a modern economy. (A bulldog can learn to ride a skateboard, but it cannot learn to make a skateboard.)

What would we think of them? Would they be “human”? Even though they look like us, they could only live in houses and wear clothes if we gave those to them; if not given food, they would have to stay in their native environment and hunt.

It is hard to imagine a “human” without technology, nor explaining the course of human history without the advance of technology. The rise of trans-Atlantic discovery, trade, and migration that so fundamentally shaped the past 500 years cannot be explained without noting the development of superior European ships, maps, and navigational techniques necessary to make the journey. Why did Europe discover America and not the other way around? Many reasons, but fundamentally, because the Americans of 1492 didn’t have ships that could make the journey and the Europeans did.

The Romans were a surprisingly advanced society, technology wise, and even during the High Middle Ages, European technology continued to advance, as with the spread of wind and water mills. But I think it was the adoption of (Hindu-)Arabic numerals, popularized among mathematicians in the 1200s by Fibonacci, but only adopted more widely around the 1400s, that really allowed the Industrial and Information Revolutions to occur. (Roman numerals are just awful for any maths.)

From the Abacus and Fibonacci’s Liber Abaci, Napier developed his “bones,” the first European mechanical calculating machine, in 1617. These were followed by Pascal’s Calculator in 1642, the slide rule (early 1600s,) and Leibniz’s Stepped Reckoner in 1672. After that, progress on the adding machine problem was so rapid that it does not do to list all of the devices and prototypes devised, but we may mention Gaspard de Prony’s impressive logarithmic and trigonometric mathematical tables, for use by human “calculators”, and Babbage‘s analytical and difference machines. The Arithmometer, patented in 1820, was the first commercially successful mechanical calculator, used in many an office for nearly a century.

The history of mechanical computing devices wouldn’t be complete without reference to the parallel development of European clocks and watches, which pioneered the use of gears to translate movement into numbers, not to mention the development of an industry capable of manufacturing small, high-quality gears to reasonably high tolerances.

Given this context, I find our culture’s focus on Babbage–whose machines, while impressive, was never actually built–and his assistant Ada of Lovelace, a bit limited. Their contributions were interesting, but taken as a whole, the history is almost an avalanche of innovations.

Along the way, the computer has absorbed many technological innovations from outside computing–the Jacquard Loom pioneered the use of punch cards; the lightbulb pioneered the vacuum tubes that eventually filled Colossus and ENIAC.

But these early computers had a problem: vacuum tubes broke often.

During and immediately after World War II a phenomenon named “the tyranny of numbers” was noticed, that is, some computational devices reached a level of complexity at which the losses from failures and downtime exceeded the expected benefits.[2] Each Boeing B-29 (put into service in 1944) carried 300–1000 vacuum tubes and tens of thousands of passive components.[notes 4] The number of vacuum tubes reached thousands in advanced computers and more than 17,000 in the ENIAC (1946).[notes 5] Each additional component reduced the reliability of a device and lengthened the troubleshooting time.[2] Traditional electronics reached a deadlock and a further development of electronic devices required reducing the number of their components.

Also:

…the 1946 ENIAC, with over 17,000 tubes, had a tube failure (which took 15 minutes to locate) on average every two days. The quality of the tubes was a factor, and the diversion of skilled people during the Second World War lowered the general quality of tubes.[29]

The invention of the semiconductor further revolutionized computing–bringing us a long way from the abacus of yesterday.

Chapter 3 takes a break from the development of beautiful machines to examine their effects on humans. Auerswald writes:

By the twentieth century, the systematic approach to analyzing the division of labor that de Prony developed would have a name: management science. the first and foremost proponent of management science was Frederick Winslow Taylor, a child of privilege who found his calling in factories. …

This first experience of factory work gave Taylor an understanding of the habits of workers that was as intimate as it was, ultimately, unfavorable. Being highly organized and precise by nature, Taylor was appalled at the lax habits and absence of structure that characterized the early twentieth-century factory floor. … However, Taylor ultimately concluded that the blame did not lie with the workers but in the lack of rigorously considered management techniques. At the center of management, Taylor determined, was the capacity to precisely define the tasks of which a “job” was comprised.

What distinguished Taylor was his absolute conviction that worker could not be left on their own to define, much less refine, the tasks that comprised their work. He argued that authority must be fully vested in scientifically determined routine–that is to say, code.

Sounds hellish.

I know very little of management science beyond what can be found in Charlie Chaplin’s Modern Times. According to Wikipedia, Vladimir Lenin described Taylorism as a “‘scientific’ system of sweating” more work from laborers.[3] However, in Taylor’s defense, I don’t think the adoption of Taylorism ever resulted in the mass starvation of millions of people, so maybe Lenin should shut up was wrong. Further:

In the course of his empirical studies, Taylor examined various kinds of manual labor. For example, most bulk materials handling was manual at the time; material handling equipment as we know it today was mostly not developed yet. He looked at shoveling in the unloading of railroad cars full of ore; lifting and carrying in the moving of iron pigs at steel mills; the manual inspection of bearing balls; and others. He discovered many concepts that were not widely accepted at the time. For example, by observing workers, he decided that labor should include rest breaks so that the worker has time to recover from fatigue, either physical (as in shoveling or lifting) or mental (as in the ball inspection case). Workers were allowed to take more rests during work, and productivity increased as a result.[11]

Also:

By factoring processes into discrete, unambiguous units, scientific management laid the groundwork for automation and offshoring, prefiguring industrial process control and numerical control in the absence of any machines that could carry it out. Taylor and his followers did not foresee this at the time; in their world, it was humans that would execute the optimized processes. (For example, although in their era the instruction “open valve A whenever pressure gauge B reads over value X” would be carried out by a human, the fact that it had been reduced to an algorithmic component paved the way for a machine to be the agent.) However, one of the common threads between their world and ours is that the agents of execution need not be “smart” to execute their tasks. In the case of computers, they are not able (yet) to be “smart” (in that sense of the word); in the case of human workers under scientific management, they were often able but were not allowed. Once the time-and-motion men had completed their studies of a particular task, the workers had very little opportunity for further thinking, experimenting, or suggestion-making. They were forced to “play dumb” most of the time, which occasionally led to revolts.

While farming has its rhythms–the cows must be milked when the cows need to be milked, and not before or after; the crops must be harvested when they are ripe–much of the farmer’s day is left to his own discretion. Whether he wants to drive fence in the morning and hoe the peas in the afternoon, or attend to the peas first and the fence later is his own business. If he wants to take a nap or pause to fish during the heat of the day it is, again, his own business.

A factory can’t work like that. If the guy who is supposed to bolt the doors onto the cars can’t just wander off to eat a sandwich and use the restroom whenever he feels like it, nor can he decide that today he feels like installing windshields. Factories and offices allow many men to work together by limiting the scope of each one’s activities.

Is this algorithmisation of labor inhuman, and should we therefore welcome its mechanization and automation?

# Why are there no Han Chinese Fields Medalists?

I am specifically referring to Han Chinese from the People’s Republic of China (hereafter simply called “China,”) but wanted to keep the title to a reasonable length.

There are about a billion Han Chinese. They make up about 90% of the PRC, and they have some of the highest average IQs on the planet, with particularly good math scores.

Of the 56 Fields Medals (essentially, the Nobel for Math) awarded since 1936, 12 (21%) have been French. 14 or 15 have been Jewish, or 25%-27%.

By contrast, 0 have been Han Chinese from China itself.

France is a country of 67.15 million people, of whom about 51 million are native French. The world has about 14-17.5 million Jews. China has about 1.37 billion people, of whom 91.51% are Han, or about 1.25 billion.

Two relatively Chinese people have won Fields medals:

Shing-Tung Yau was born in China, but is of Hakka ancestry (the Hakka are an Asian “market-dominant minority,”) not Han. His parents moved to Hong Kong when he was a baby; after graduating from the Chinese University of Hong Kong, he moved to the US, where he received his PhD from Berkley. Yau was a citizen of British-owned Hong Kong (not the People’s Republic of China), when he won the Fields Medal, in 1982; today he holds American citizenship.

Terence Tao, the 2006 recipient, is probably Han (Wikipedia does not list his ethnicity.) His father hailed from Shanghai, China, but moved to Hong Kong, where he graduated from medical school and met Tao’s mother, another Hong Kong-ian. Tao himself was born in Australia and later moved to the US. (Tao appears to be a dual Australian-American citizen.)

(With only 7.4 million people, Hong Kong is doing pretty well for itself in terms of Fields Medalists with some form of HK ancestry or citizenship.)

Since not many Fields Medals have been awarded, it is understandable why the citizens of small countries, even very bright ones, like Singapore, might not have any. It’s also understandable why top talent often migrates to places like Hong Kong, Australia, or the US. But China is a huge country with a massive pool of incredibly smart people–just look at Shanghai’s PISA scores. Surely Beijing has at least a dozen universities filled with math geniuses.

So where are they?

Is it a matter of funding? Has China chosen to funnel its best mathematicians into applied work? A matter of translation? Does the Fields Medal Committee have trouble reading papers written in Chinese? A matter of time? Did China’s citizens simply spent too much of the of the past century struggling at the edge of starvation to send a bunch of kids off to university to study math, and only recently achieved the level of mass prosperity necessary to start on the Fields path?

Whatever the causes of current under-representation, I have no doubt the next century will show an explosion in Han Chinese mathematical accomplishments.

# Homeschooling Corner: A Mathematician’s Lament, by Paul Lockhart

Paul Lockhart’s A Mathematician’s Lament: How School Cheats us of our Most Fascinating and Imaginative Artform is a short but valuable book, easily finished in an afternoon.

Lockhart’s basic take is that most of us have math backwards. We approach (and thus teach) it as useful but not fun–something to be slogged through, memorized, and then avoided as much as possible. By contrast, Lockhart sees math as more fun than useful.

I do not mean that Lockhart denies the utility of balancing your checkbook or calculating how much power your electrical grid can handle, but most of the math actual mathematicians do isn’t practical. They do it because they enjoy it; they love making patterns with numbers and shapes. Just because paint has a very practical use in covering houses doesn’t mean we shouldn’t encourage kids to enjoy painting pictures; similarly, Lockhart wants kids to see mathematics as fun.

But wait, you say, what if this loosey-goosey, free-form, new math approach results in kids who spend a lot of time trying to re-derive pi from first principles but never really learning algebra? Lockhart would probably counter that most kids never truly master algebra anyway, so why make them hate it in the process? Should we only let kids who can paint like the Masters take art class?

If you and your kids already enjoy math, Lockhart may just reinforce what you already know, but if you’re struggling or math is a bore and a chore, Lockhart’s perspective may be just what you need to turn things around and make math fun.

For example: There are multiple ways to group the numbers during double-digit multiplication, all equally “correct”; the method you chose is generally influenced by things like your familiarity with double-digit multiplication and the difficulty of the problem. When I observed one of my kids making errors in multiplication because of incorrect regrouping, I showed them how to use a more expanded way of writing out the numbers to make the math clearer–promptly eliciting protests that I was “doing it wrong.” Inspired by Lockhart, I explained that “There is no one way to do math. Math is the art of figuring out answers, and there are many ways to get from here to there.” Learning how to use a particular approach—“Put the numbers here, here, and here and then add them”–is useful, but should not be elevated above using whatever approach best helps the child understand the numbers and calculate the correct answers.

The only difficulty with Lockhart’s approach is figuring out what to actually do when you sit down at the kitchen table with your kids, pencil and paper in hand. The book has a couple of sample lessons but isn’t a full k-12 curriculum. It’s easy to say, “I’m going to do a free-form curriculum that requires going to the library every day and uses every experience as a learning opportunity,” and rather harder to actually do it. With a set curriculum, you at least know, “Here’s what we’re going to do today.”

My own personal philosophy is that school time should be about 50% formal instruction and 50% open-ended exploration. Kids need someone to explain how the alphabet works and what these funny symbols on the math worksheet mean; they also need time to read fun books and play with numbers. They should memorize their times tables, but a good game can make times tables fun. In short, I think kids should have both a formal, straightforward curriculum or set of workbooks (I have not read enough math textbooks to recommend any particular ones,) and a set of math enrichment activities, like tangrams, pattern blocks, reading about Penrose the Mathematical Cat, or watching Numberphile on YouTube.

(Speaking of Penrose, I thought the chapter on binary went right over my kids’ heads, but yesterday they returned all of their answers in math class in binary, so I guess they picked up more than I gave them credit for.)

YouCubed.org is an interesting website I recently discovered. So far we’ve only done two of the activities, but they were cute and I suspect the website will make a useful addition to our lessons. If you’ve used it, I’d love to hear your thoughts on it.

That’s all for now. Happy learning!

# Homeschooling Corner: Math ideas and manipulatives for younger grades

When you love a subject and your kids love it, too, it’s easy to teach. When you’re really not sure how to approach the subject or your kids hate it, it gets a lot trickier. (See: spelling.)

So I thought I’d make a list of some of our favorite math related materials–but please remember, all you really need for teaching math is a paper and pencil. (Or less–Archimedes did math with a stick and some sand!)

Countables

Little ones who are just learning to count and add benefit from having something concrete they can hold, touch, and move around when thinking about concepts like “two more” or “two less.”

You can count almost anything–pebbles, shells, acorns, pennies, Monopoly money, fingers–but having a box of dedicated, fun, colorful countables on hand is useful. My favorites:

Abacus. The abacus has the lovely advantage that all of its counters are on rods and so don’t get scattered around the room, stepped on and lost. I made my own abacus (inspired by commenter Dave‘s abacus) out of a shoe box, plastic beads, pipecleaners, and tape. You can count, add, subtract, multiply, divide, etc., on an abacus, but for your purposes you’ll just need to learn addition and subtraction.

Different abaci have different numbers and arrangements of beads. If your kids are still learning to count/mastering addition and subtraction up to ten (standard kindergarten goals,) I’d use an abacus with 9 beads per string. (Just like writing numbers, after you get to nine on the “ones” string, you raise up one bead on the “10” string.)

We adults tend to take place value for granted (“it’s obvious that we use the decimal system!”) but switching from column to column can be confusing for young kids. There’s no intuitive reason why 11 doesn’t = 2. The abacus helps increase awareness of place value (typically taught in first grade) because you simply run out of beads after 9 and have to switch to the next row.

Once kids have the basic idea, you can switch to a more advanced abacus like the Soroban. The top bead on the Soroban is worth 5, so students count 1-2-3-4, then click the 5 bead and clear the unit beads, then add unit beads to the five to count 6-7-8-9, then click one bead in the tens column and clear all of the beads in the unit and five column. My apologies if it sounds complicated; it really isn’t, it’s just a little tricky to put into words.

You can get abacus workbooks; I have not used any so I cannot review them but they look fun. Rather, I just use the abacus as a complement to the other math problems we are already doing. (I have read Mr. Green’s How to Use a Chinese Abacus, which was the only book my library had on the subject. It is a very good introduction aimed at adults.)

Counting Penguins

There is nothing magical about penguins; I just happen to like them. The set has 100 penguins in ten sets (distinguished by color) plus ten “ice bars” that hold ten penguins each. (Besides addition and subtraction,) I find these useful for introducing and visualizing multiplication , eg, 3 rows of 5 penguins = 3×5.

Counting Cubes

For bigger numbers, we have a bag of 1,000 interlocking cubes. Kids will want to just plain build with them, like Legos, which is fine–a fun treat after hard work. You can easily use these to represent 1s, 10s, and 100s (it takes a while to assemble a full 1,000 cube,) and to represent operations like 3x3x3, helping bridge both place value and multiplication. Legos work for this, too, though you’ll probably want to sort out ones that are all the same size and shape.

Pattern Blocks

(I think I’ve been incorrectly calling these tangrams, though the principles are similar.)

These pattern blocks are a family heirloom, sent to me by my grandmother upon the birth of my first child. I played with them when I was a child; my siblings played with them; now my children play with them. Someday I will pass them on to my grandchildren… but you can also get them on Amazon. (We use these with a book of pattern block activities that hails from the 80s; I am sure there are many good books of a similar nature published within the past couple of decades.

Apparently there are workbooks with pattern block activities aimed all the way up to 8th grade, but I have not read them and cannot comment on them.

Cuisenaire rods

We didn’t use cuisenaire rods when I was young, but I think I would have liked them. Similar to the tangrams pattern blocks, there are lots of interesting workbooks, games, and other activities you can do with these.

Building toys

Open-ended building toys (Legos, Tinker Toys, blocks, magnetic tiles) come in almost endless forms and can be used to build all sorts of geometric shapes.

Fractions

Fraction blocks and fraction circles are both handy.

Games

Almost any kids’ board game can be transformed into a math game by adding cards with math problems to be solved before completing a turn or using math dice. Your local games shop can help you find dice with numbers higher than six, or you can just tape paper onto an existing cube to make a custom die of your liking (like an + and – die). There are also tons of fun logic games; I pull these out whenever kids start getting restless.

Books

There are so many great math books, from Sir Cumference to Penrose, that I can’t hope to list them all. I encourage you to check out your library’s selection. Here are a few of my favorites:

The Adventures of Penrose the Mathematical Cat (plus sequels) makes a very pleasant enrichment portion of our daily maths. Each day we read one of Penrose’s stories (on subjects like Fibonacci numbers, primes, operations, etc) and do a short, related math activity.

Penrose is probably most appropriate for kids in mid to late elementary, not little ones just learning to count and add. (Note: the first story in the book was about binary, which flew over my kids’ heads.) Sir Cumference is more appropriate for younger learners.

Mathematicians are People, Too: biographies of great mathematicians. I’m not keen on the title, but my kids liked the chapter on Archimedes.

Balance Benders These workbooks come in different levels, from beginner to expert. Each puzzle presents students with a drawing of a balance with shapes on either side, and asks them to figure out, from a choice of answers, which statements about the shapes are true, eg “One circle equals two squares” after viewing a balance with two circles and four squares. (We also do logic puzzles and picture sudoku.)

Textbooks

I am not recommending any textbooks because I don’t have any idea which is the best. We don’t use a pre-packaged curriculum, because they tend to be expensive–instead I’ve just picked up a whole bunch of different math texts at the second hand shop and been gifted some lovely hand-me-downs from relatives. At this point I might have too many math books… I use 3 or 4 interchangably, depending on exactly which concepts we’re covering and whether I think the kids need more practice or not. I recently lucked into a volume of the “What your X Grader Needs to Know” series, and it gives a very nice overview of grade-level math expectations (among other things.)

Incidentally, the local public school math expectations appear to be:

Kindergarten: Reliably add and subtract the numbers 0-10; add small numbers to numbers between 10 and 20; be able to write all of the numbers from 0-20; count to 100.

1st grade: Place value; add and subtract one and two digit numbers with no regrouping.